Simulation method and simulation device

ABSTRACT

A state of a particle which was in a liquid state at a first time is calculated at a second time after the first time when a continuum including a liquid and a solid is represented by the plurality of particles. It is determined whether the particle has become a first solid particle at the second time. The first solid particle and all particles belonging to a solid which includes a second solid particle arranged in a predetermined range from the first solid particle are defined as particles belonging to the same solid when it is determined that the particle which was in the liquid state at the first time has become the first solid particle at the second time. The state of each of the particles belonging to the same solid is calculated using an equation of motion of a rigid body.

CROSS-REFERENCE TO RELATED APPLICATION(S)

This application is based upon and claims the benefit of priority of theprior Japanese Patent Application No. 2013-016206, filed on Jan. 30,2013, the entire contents of which are incorporated herein by reference.

FIELD

The embodiments discussed herein are related to a simulation method anda simulation device.

BACKGROUND

As a numerical calculation method of calculating the motion of acontinuum such as a fluid or an elastic body, for example, a finitedifference method, a finite element method, or a finite volume methodhas been used which finds the approximate solution of a differentialequation on the basis of the numerical mesh. In addition, in recentyears, since numerical calculation has been used in the field ofapplication such as computer aided engineering (CAE), the numericalcalculation method of calculating the state of the continuum has beendeveloped and the problem of the interaction between a fluid and astructure has been solved. However, in the numerical calculation methodusing the numerical mesh, when a moving boundary problem, such as theexistence of an interface including a free surface or a problem influid-structure interaction analysis for analyzing the interactionbetween a fluid and a structure, occurs, handling of the continuumbecomes complicated. Therefore, in some cases, it is difficult to createa program.

As the numerical calculation method without using the numerical mesh,there is a particle method. The particle method analyzes the motion of acontinuum as the motion of a finite number of particles. Arepresentative particle method which is currently proposed is, forexample, a smoothed particles hydrodynamics (SPH) method or a movingparticles semi-implicit (MPS) method. The particle method can analyzethe motion of the continuum without a special measure in the treatmentof the moving boundary. Therefore, in recent years, the particle methodhas been widely used as the numerical calculation method of calculatingthe motion of the continuum.

In particular, in the pressing of metal such as casting or forging,metal is processed through a complicated process. For example, metal(solidified metal) which is cooled and solidified is mixed with liquidmetal, the solidified metal is grown, and the volume of metal is changedin the solidification process. The particle method is expected to beactively used in casting and forging simulations since the particlemethod has the advantage that it is easy to treat the free surface, itis relatively easy to calculate a parallel performance and interactionwith a solid, or the like.

Cleary method has been known as a method of calculating a process(solidification process) in which a liquid is cooled and solidified,which is a basic technique for simulating a casting process. The Clearymethod calculates the time evolution of the internal energy of eachliquid particle using the SPH method which is one of the particlemethods and calculates the temperature, density, and viscositycoefficient of the liquid particle as a function of internal energy.That is, when the internal energy is reduced and the temperature islowered, the Cleary method increases the viscosity coefficient of theliquid to represent solidification and increases the density of theliquid to represent a reduction in volume due to solidification.

The Cleary method discretizes the equation of a fluid using the SPHmethod as represented by the following Expressions (1) to (4):

$\begin{matrix}{\frac{\mathbb{d}\rho_{i}}{\mathbb{d}t} = {\sum\limits_{j}^{\;}{{m_{j}\left( {v_{i} - v_{j}} \right)} \cdot \frac{\partial{W\left( {{x_{i} - x_{j}}} \right)}}{\partial x_{i}}}}} & (1) \\{\frac{\mathbb{d}v_{i}}{\mathbb{d}t} = {g - {\sum\limits_{j}^{\;}{{m_{j}\left\lbrack {\left( \frac{\rho_{j} + \rho_{i}}{\rho_{j}\rho_{i}} \right) - {\frac{\xi}{\rho_{j}\rho_{i}}\frac{4\mu_{i}\mu_{j}}{\left( {\mu_{i} + \mu_{j}} \right)}\frac{v_{ij} \cdot x_{ij}}{{x_{ij}}^{2} + \eta^{2}}}} \right\rbrack}\frac{\partial{W\left( {{x_{i} - x_{j}}} \right)}}{\partial x_{i}}}}}} & (2) \\{p_{i} = {P_{0}\left\lbrack {\left( \frac{\rho_{i}}{\rho_{s,i}} \right)^{\gamma} - 1} \right\rbrack}} & (3) \\{\frac{\mathbb{d}U_{i}}{\mathbb{d}t} = {\sum\limits_{j}^{\;}{\frac{4m_{j}}{\rho_{j}\rho_{i}}\frac{k_{i}k_{j}}{\left( {k_{i} + k_{j}} \right)}{\frac{x_{ij}}{{x_{ij}}^{2} + \eta^{2}} \cdot \frac{\partial{W\left( {{x_{i} - x_{j}}} \right)}}{\partial x_{i}}}}}} & (4)\end{matrix}$

Expression (1) indicates the law of conservation of mass, Expression (2)indicates the law of conservation of momentum, Expression (3) indicatesa state equation, and Expression (4) indicates the law of conservationof energy. In Expressions (1) to (4), x_(i), v_(i), ρ_(i), m_(i), p_(i),and U_(i) are the position vector of a particle i, the velocity vectorof the particle i, the density of the particle i, the mass of theparticle i, the pressure of the particle i, and the internal energy ofthe particle i, respectively. In addition, x_(ij) and v_(ij) are therelative position vector and relative velocity vector of particles i andj, respectively, and x_(ij)=x_(i)−x_(j) and v_(ij)=v_(i)−v_(j) areestablished. Furthermore, κ_(i) and μ_(i) are the thermal conductivityof the particle i and the viscosity coefficient of the particle i,respectively. In addition, P₀=ρ₀c² is established and c is the speed ofsound. Further, ρ_(s,i) is the reference density of the particle i andpressure is 0 when ρ_(i)=ρ_(s,i) is established.

In addition, W is a kernel function and, for example, a spline functionrepresented by the following Expression (5) is used as W.

$\begin{matrix}{{W\left( {r,h} \right)} = \left\{ \begin{matrix}{\left( {1 - {1.5\left( \frac{r}{h} \right)^{2}} + {0.75\left( \frac{r}{h} \right)^{3}}} \right)/\beta} & {{0 \leq \frac{r}{h} \leq 1},} \\{0.25{\left( {2 - \frac{r}{h}} \right)^{3}/\beta}} & {{1 \leq \frac{r}{h} \leq 2},} \\0 & {2 \leq {\frac{r}{h}.}}\end{matrix} \right.} & (5)\end{matrix}$

In Expression (5), h is an influence radius between particles. Forexample, as h, a value that is about two to three times that the averagedistance between the particles in the initial state is used. Inaddition, β is a value which is adjusted such that the entire spaceintegration amount of the kernel function is 1. In the case of twodimensions, β is set to 0.7 πh². In the case of three dimensions, β isset to πh³.

In the Cleary method, when the internal energy is reduced and thetemperature is lower than a melting point, the viscosity coefficientμ_(i) is increased and the effect of canceling the relative velocitybetween the particles represented by the third term of Expression (2) isimproved. Therefore, it is difficult to deform by the third term. Inthis way, the Cleary method represents solidification. In addition, inthe Cleary method, when the reference density ρ_(s,i) increases,pressure is reduced and the surrounding particles are collected by theeffect of the second term of Expression (2). In this way, the Clearymethod represents contraction due to solidification.

It is possible to perform a simulation by calculating the time evolutionof Expressions (1) to (4) using the Euler's method or the Leapfrogmethod which is a general ordinary differential equation.

In the Cleary method, since the value of the viscosity coefficientincreases in the solidification process, a time step is very small incalculation. Therefore, the number of calculation operations increasesuntil calculation ends. As a result, the Cleary method has a longcalculation time.

As an example of a method of calculating the interaction between a fluidand a rigid body, there is a method which uses the equation of motion ofa liquid for a liquid portion and uses the equation of motion of a rigidbody for a solid portion. In the method, since the motion of the solidportion is calculated by the equation of motion of a rigid body, thecalculation time is shorter than that in the Cleary method.

As to the conventional techniques, refer to Paul W. Cleary, “Extensionof SPH to predict feeding freezing and defect creation in low pressuredie casting”, Applied Mathematical Modeling, 34 (2010), pp. 3189-3201;and Koshizuka, S., Nobe A. and Oka Y. “Numerical Analysis of BreakingWaves Using the Moving Particle Semi-implicit Method”, Int. J. Numer.Meth. Fluids, 26, 751-769 (1998), for example.

However, in the method which uses the equation of motion of a rigid bodyfor the solid portion, the accuracy of the calculation result is nothigh in a situation in which a new solid is generated from a liquid. Forexample, a case in which liquid metal is poured into a mold and thencooled will be described. In this case, a plurality of portions of theliquid metal starts to be solidified depending on the cooling conditionsand the volume of the plurality of solidified portions increases overtime. Then, the entire liquid metal is solidified. FIG. 14 is a diagramillustrating an example of the problems of the method according to therelated art. In the example illustrated in FIG. 14, particles 90 a of asolid portion 90, particles 91 a of a solid portion 91, and particles 92a of a liquid portion 92 in the metal which is solidified by cooling arepresent in a mold. In this case, even when the solidified volume of thesolid portion 90 is increased, the liquid portion 92 is solidified, andthe solidified portion 92 and the solid portion 90 form the same solidby cooling, the above-mentioned method treats the solidified portion 92and the solid portion 90 as individual solids. That is, theabove-mentioned method separately calculates the motion of thesolidified portion 92 and the motion of the solid portion 90 using theequation of motion of a rigid body. Therefore, the above-mentionedmethod separately calculates the motions of a plurality of solids eventhough there is originally one solid. As a result, the accuracy of thecalculation result is not high.

SUMMARY

According to an aspect of an embodiment, a simulation method causes acomputer to perform: calculating a state of a particle which was in aliquid state at a first time among a plurality of particles at a secondtime after the first time when a continuum including a liquid and asolid is represented by the plurality of particles; determining whetherthe particle which was in the liquid state at the first time has becomea first solid particle at the second time on the basis of the state ofthe particle at the second time; defining the first solid particle andall particles belonging to a solid which includes a second solidparticle arranged in a predetermined range from the first solid particleas particles belonging to the same solid when it is determined that theparticle which was in the liquid state at the first time has become thefirst solid particle at the second time; and calculating the state ofeach of the particles belonging to the same solid using an equation ofmotion of a rigid body.

According to another aspect of an embodiment, a simulation deviceincludes a calculation unit, a determination unit, and a definitionunit. The calculation unit calculates a state of a particle which was aliquid state at a first time among a plurality of particles at a secondtime after the first time when a continuum including a liquid and asolid is represented by the plurality of particles, and calculates thestate of each particle belonging to the defined same solid using anequation of motion of a rigid body. The determination unit determineswhether the particle which was in the liquid state at the first time hasbecome a first solid particle at the second time on the basis of thestate of the particle at the second time. The definition unit definesthe first solid particle and all particles belonging to a solid whichincludes a second solid particle arranged in a predetermined range fromthe first solid particle as particles belonging to the same solid whenit is determined that the particle which was in the liquid state at thefirst time has become the first solid particle at the second time.

The object and advantages of the invention will be realized and attainedby means of the elements and combinations particularly pointed out inthe claims.

It is to be understood that both the foregoing general description andthe following detailed description are exemplary and explanatory and arenot restrictive of the invention, as claimed.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a diagram illustrating an example of a process performed by asimulation device according to an embodiment;

FIG. 2 is a diagram illustrating an example of the functional structureof the simulation device according to the embodiment;

FIG. 3 is a diagram illustrating an example of a metal model indicatedby metal model data;

FIG. 4 is a diagram illustrating an example of the metal model indicatedby the metal model data;

FIG. 5 is a diagram illustrating an example of the process performed bythe simulation device according to the embodiment;

FIG. 6 is a diagram illustrating an example of the process performed bythe simulation device according to the embodiment;

FIG. 7 is a diagram illustrating an example of the process performed bythe simulation device according to the embodiment;

FIG. 8 is a diagram illustrating an example of the process performed bythe simulation device according to the embodiment;

FIG. 9 is a diagram illustrating an example of the process performed bythe simulation device according to the embodiment;

FIG. 10 is a diagram illustrating an example of the process performed bythe simulation device according to the embodiment;

FIGS. 11A and 11B are flowcharts illustrating the procedure of asimulation process according to the embodiment;

FIG. 12 is a flowchart illustrating the procedure of a solid numberdefinition process according to the embodiment;

FIG. 13 is a diagram illustrating a computer which executes a simulationprogram; and

FIG. 14 is a diagram illustrating an example of the problems of a methodaccording to the related art.

DESCRIPTION OF EMBODIMENTS

Preferred embodiments of the present invention will be explained withreference to accompanying drawings. The embodiments do not limit thedisclosed technique.

Structure of Simulation Device

The simulation device according to the embodiment will be described. Thesimulation device according to this embodiment calculates the state ofeach particle of a continuum including a metallic liquid and a metallicsolid at each time step t_(ts) using a particle method, according to ascenario which pours the metallic liquid into a mold and cools themetallic liquid. FIG. 1 is a diagram illustrating an example of aprocess performed by the simulation device according to the embodiment.As illustrated in FIG. 1, the simulation device pours a metallic liquid20 into a mold 21, cools the metallic liquid 20, and calculates thestate of each particle when the metallic liquid 20 and a metallic solid22, which is a solidified metallic liquid 20, are mixed with each otherusing the particle method.

FIG. 2 is a diagram illustrating an example of the functional structureof the simulation device according to the embodiment. As illustrated inFIG. 2, a simulation device 10 includes an input unit 11, a display unit12, a storage unit 13, and a control unit 14.

The input unit 11 inputs information to the control unit 14. Forexample, the input unit 11 receives a simulation execution instructionto perform a simulation process, which will be described below, from theuser and inputs the received simulation execution instruction to thecontrol unit 14. In addition, the input unit 11 receives the initialvalue of each particle in an initial state from the user and inputs thereceived initial value of each particle to the control unit 14. Theinitial value of each particle in the initial state includes theposition, density, velocity, internal energy, state, and solid number ofeach particle. When the internal energy is greater than a predeterminedvalue and the temperature is higher than a melting point, the particleis a liquid. Therefore, information indicating that the particle is aliquid is set to the state of the particle. In addition, when theinternal energy is equal to or less than the predetermined value and thetemperature is lower than a solidifying point, the particle is a solid.Therefore, information indicating that the particle is a sold is set tothe state of the particle. An identification number of the solidincluding the particle is set to the solid number. An exemplary deviceof the input unit 11 is a keyboard or a mouse.

The display unit 12 displays various kinds of information. For example,the display unit 12 displays a simulation result under the control of adisplay control unit 14 e, which will be described below. An exemplarydevice of the display unit 12 is a liquid crystal display.

The storage unit 13 stores various programs executed by the control unit14. In addition, the storage unit 13 stores metal model data 13 a. Themetal model data 13 a indicates a metal model in which a continuumincluding a metallic liquid and a metallic solid is represented as aplurality of particles. FIGS. 3 and 4 are diagrams illustrating anexample of the metal model indicated by the metal model data. In theexample illustrated in FIG. 3, a portion of the metallic liquid 20poured into the mold 21 is solidified, and the metallic liquid 20 andthe metallic solid 22 are mixed with each other. The metal model of aportion 23 illustrated in FIG. 3 will be described in detail. FIG. 4 isa diagram illustrating details of the metal model of the portion 23illustrated in FIG. 3. As illustrated in the example of FIG. 4, themodel of the metallic liquid 20 and the metallic solid 22 includes aplurality of particles 25. In this embodiment, the position, density,velocity, internal energy, state, and solid number of each particle 25are calculated at each time step t_(ts).

Returning to FIG. 1, the storage unit 13 is a semiconductor memorydevice such as a flash memory, or a storage device such as a hard diskor an optical disk. The storage unit 13 is not limited to the abovetypes of storage devices, but may be a random access memory (RAM) or aread only memory (ROM).

The control unit 14 includes an internal memory for storing a program orcontrol data which defines various types of procedures, and varioustypes of processes are performed by the program or control data. Asillustrated in FIG. 2, the control unit 14 includes a calculation unit14 a, an update unit 14 b, a determination unit 14 c, a definition unit14 d, and the display control unit 14 e.

The calculation unit 14 a calculates various kinds of information. Forexample, the calculation unit 14 a performs time evolution calculationusing the equation of motion of a fluid for a liquid particle, which isa particle in a liquid state, at each time step t_(ts) to calculate theposition, velocity, density, and internal energy of the liquid particle.For example, the calculation unit 14 a calculates the position,velocity, density, and internal energy of all particles, which were inthe liquid state at a time step (t_(ts)−1), at the time step t_(ts)using the above-mentioned Expressions (1) to (4).

The calculation unit 14 a performs time evolution calculation using theequation of motion of a rigid body for a solid including a solidparticle, which is a particle in a solid state, at each time step t_(ts)to calculate the position, velocity, density, and internal energy of thesolid particle. For example, the calculation unit 14 a calculates thetranslation motion of the gravity center of the particle, which is inthe solid state at the time step (t_(ts)−1), and the rotational motionof the solid particle about the gravity center at the time step t_(ts).In this way, the calculation unit 14 a calculates the time evolution ofthe solid including solid particles at the time step t_(ts). That is,the calculation unit 14 a calculates the translation motion of thegravity center of each solid and the rotational motion of the solidabout the gravity center at the time step t_(ts) from the force appliedto the solid particles in each solid. Then, the calculation unit 14 acalculates the position, velocity, and density of the solid particles ineach solid at the time step t_(ts) from the calculated translationmotion of the gravity center of the solid and the calculated rotationalmotion of the solid about the gravity center, using the equation ofmotion of a rigid body. In addition, the calculation unit 14 acalculates the internal energy of the solid particles at the time stept_(ts) using the above-mentioned Expression (4).

An aspect of the calculation unit 14 a will be described. For example,when the simulation execution instruction is input from the input unit11, first, the calculation unit 14 a sets the value of the time stept_(ts) to 0. Then, the calculation unit 14 a determines whether theinitial value of each particle is input from the input unit 11. When theinitial value is input, the calculation unit 14 a increases the value ofthe time step t_(ts) by 1. In addition, when the display control unit 14e determines that the value of the time step t_(ts) is equal to or lessthan the last time step N_(L) of the simulation, the calculation unit 14a increases the value of the time step t_(ts) by 1.

Then, the calculation unit 14 a calculates the position, velocity,density, and internal energy of all particles, which were in the liquidstate at the time step (t_(ts)−1), at the time step t_(ts) using theabove-mentioned Expressions (1) to (4).

In addition, the calculation unit 14 a performs time evolutioncalculation using the equation of motion of a rigid body for all solidsincluding the particles which are in the solid state at the time step(t_(ts)−1) and performs the next process. That is, the calculation unit14 a calculates the position, velocity, density, and internal energy ofall particles, which were in the solid state at the time step(t_(ts)−1), at the time step t_(ts).

The update unit 14 b updates various kinds of information. An aspect ofthe update unit 14 b will be described. For example, the update unit 14b updates the position, velocity, density, and internal energy of allparticles which were in the liquid state at the time step (t_(ts)−1) tothe calculated position, velocity, density, and internal energy of theliquid particles at the time step t_(ts), respectively.

In addition, the update unit 14 b updates the position, velocity,density, and internal energy of all particles which were in a solidstate at the time step (t_(ts)−1) to the calculated position, velocity,density, and internal energy of the solid particles at the time stept_(ts), respectively.

Then, the update unit 14 b updates the state of all particles on thebasis of the calculated internal energy. For example, when thecalculated internal energy is greater than a predetermined value, theupdate unit 14 b sets information indicating that the particle is aliquid to the state of particles to update the state. When the internalenergy is equal to or less than the predetermined value, the update unit14 b sets information indicating that the particle is a solid to thestate of particles to update the state.

Then, the update unit 14 b stores the update result (the position,velocity, density, internal energy, and state) of all particles in apredetermined area of the storage unit 13 so as to be associated withthe time step t_(ts).

Then, the determination unit 14 c performs various determinationoperations. An aspect of the determination unit 14 c will be described.For example, the determination unit 14 c determines whether there is anewly solidified particle (a particle whose state has been changed froma liquid to a solid) on the basis of the states of all particles at thetime step t_(ts) before and after the update.

When there is a newly solidified particle, the determination unit 14 cspecifies the newly solidified particle. Then, the determination unit 14c sets the solid number of the specified solid particle to an undefinedstate.

The determination unit 14 c determines whether there is a newly moltenliquid particle (a particle whose state has been changed from a solid toa liquid) on the basis of the states of all particles at the time stept_(ts) before and after the update.

When there is a newly molten liquid particle, the determination unit 14c specifies the newly molten liquid particle. Then, the determinationunit 14 c sets the solid numbers of all solid particles included in thesolid when the specified liquid particle has been a solid particle atthe time step (t_(ts)−1) to an undefined state.

Then, the determination unit 14 c determines whether there is a newlysolidified particle and there is a newly molten liquid particle on thebasis of the states of all particles at the time step t_(ts) before andafter the update.

The definition unit 14 d defines various kinds of information. An aspectof the definition unit 14 d will be described. For example, when thedetermination unit 14 c determines that there is a newly solidifiedparticle or determines that there is a newly molten liquid particle, thedefinition unit 14 d performs the next process. That is, the definitionunit 14 d determines whether there is a solid particle which has notbeen selected among the solid particles with the undefined solidnumbers.

When there is a solid particle which has not been selected, thedefinition unit 14 d selects one of the solid particles which have notbeen selected and have the undefined solid numbers. Then, the definitionunit 14 d determines whether a solid particle belonging to a solid ispresent in a sphere with a radius h which has the selected solidparticle as its center. As an example of a method of determining thesolid particles belonging to a solid, there is a method which determineswhether solid identification numbers are set to the solid numbers of thesolid particles in the sphere. In the method, when the identificationnumber is set, it is determined that the solid particle belongs to asolid. When the identification number is not set, it is determined thatthe solid particle does not belong to a solid. In addition, the radius hmay have any value. For example, the influence radius of a particle inthe particle method can be used.

FIGS. 5 to 7 are diagrams illustrating an example of the processperformed by the simulation device according to the embodiment. In theexample illustrated in FIG. 5, the definition unit 14 d selects a solidparticle 30 with an undefined solid number. In addition, in the exampleillustrated in FIG. 5, the distance between the solid particle 30 and asolid particle 31 a closest to the solid particle 30 among the solidparticles 31 a belonging to a solid 31 is r₁. In the example illustratedin FIG. 5, the distance between the solid particle 30 and a solidparticle 32 a closest to the solid particle 30 among the solid particles32 a belonging to a solid 32 is r₂. In the example illustrated in FIG.5, r₁ and r₂ are both greater than the radius h. In the exampleillustrated in FIG. 5, the definition unit 14 d determines that a solidparticle belonging to a solid is absent in the sphere with the radius hwhich has the selected solid particle 30 as its center.

In the example illustrated in FIG. 6, the definition unit 14 d selects asolid particle 35 with an undefined solid number. In the exampleillustrated in FIG. 6, the distance between the solid particle 35 and asolid particle 36 a closest to the solid particle 35 among the solidparticles 36 a belonging to a solid 36 is r₃. In the example illustratedin FIG. 6, the distance between the solid particle 35 and a solidparticle 37 a closest to the solid particle 35 among the solid particles37 a belonging to a solid 37 is r₄. In the example illustrated in FIG.6, r₃ is equal to or less than the radius h and r₄ is greater than theradius h. In the example illustrated in FIG. 6, the definition unit 14 ddetermines whether the solid particle 36 a belonging to the solid 36 ispresent in the sphere with the radius h which has the selected solidparticle 35 as its center.

In the example illustrated in FIG. 7, the definition unit 14 d selects asolid particle 40 with an undefined solid number. In the exampleillustrated in FIG. 7, the distance between the solid particle 40 and asolid particle 41 a closest to the solid particle 40 among the solidparticles 41 a belonging to a solid 41 is r₅. In the example illustratedin FIG. 7, the distance between the solid particle 40 and a solidparticle 42 a closest to the solid particle 40 among the solid particles42 a belonging to a solid 42 is r₆. In the example illustrated in FIG.7, r₅ and r₆ both are equal to or less than the radius h. In the exampleillustrated in FIG. 7, the definition unit 14 d determines that thesolid particle 41 a and the solid particle 42 a which respectivelybelong to a plurality of solids 41 and 42 are present in the sphere withthe radius h which has the selected solid particle 40 as its center.

When there is a solid particle, the definition unit 14 d determineswhether the solid particle which is determined to be present belongs toeach of the plurality of solids. For example, in the example illustratedin FIG. 6, the definition unit 14 d determines that the solid particlewhich is determined to be present does not belong to any one of theplurality of solids. In the example illustrated in FIG. 7, thedefinition unit 14 d determines that the solid particle which isdetermined to be present belongs to each of the plurality of solids.

When the solid particle which is determined to be present does notbelong to any one of the plurality of solids, the definition unit 14 dsets the value of the solid number of the solid particle in the sphereto the solid number of the selected solid particle. For example, in theexample illustrated in FIG. 6, the definition unit 14 d sets the valueof the solid number ‘36’ of the solid particle in the sphere to thesolid number of the selected solid particle 35. In this way, theparticle belonging to one solid is defined so as to belong to one solid.

On the other hand, when the solid particle which is determined to bepresent belongs to each of the plurality of solids, the definition unit14 d sets the value of the solid number of the solid particle which isclosest to the selected solid particle among a plurality of solidparticles in the sphere to the solid number of the selected solidparticle. For example, in the example illustrated in FIG. 7, thedefinition unit 14 d sets the value ‘41’ of the solid number of thesolid particle 41 a which is closest to the selected solid particle 40among the plurality of solid particles 41 a and 42 a in the sphere tothe solid number of the selected solid particle 40. In this way, theparticle belonging to one solid is defined so as to belong to one solid.

The definition unit 14 d specifies the solid particle with a solidnumber to which a value other than the value of the solid number set tothe selected solid particle is set, among the plurality of solidparticles in the sphere. For example, in the example illustrated in FIG.7, the definition unit 14 d specifies the solid particle 42 a with asolid number to which a value ‘42’ other than the value ‘41’ of thesolid number set to the selected solid particle is set, among theplurality of solid particles 41 a and 42 a in the sphere.

Then, the definition unit 14 d updates the values of the solid numbersof all solid particles belonging to the solid including the specifiedsolid particle to the value set to the solid number of the selectedsolid particle. For example, in the example illustrated in FIG. 7, thedefinition unit 14 d updates the values of the solid numbers of allsolid particles 42 a (three solid particles 42 a) belonging to the solid42 including the specified solid particle 42 a to the value ‘41’ set tothe solid number of the selected solid particle 40. In this way, thesolid particles belonging to a plurality of solids are defined so as tobelong to one solid through the selected solid particle.

When a solid particle belonging to a solid is absent in the sphere withthe radius h which has the selected solid particle as its center, thedefinition unit 14 d sets, to the solid number of the selected solidparticle, a value which does not overlap the values of the solid numbersof the other solid particles. For example, in the example illustrated inFIG. 5, the definition unit 14 d sets, to the solid number of theselected solid particle 30, a value ‘50’ which does not overlap thevalues of the solid numbers of the other solid particles.

Then, the definition unit 14 d performs the process subsequent to theprocess of determining whether there is a solid particle which has notbeen selected among the solid particles with the undefined solid numbersagain. Therefore, the definition unit 14 d can set the solid numbers ofall solid particles with the undefined solid numbers. Then, when thereis no solid particle which has not been selected among the solidparticles with the undefined solid numbers, the definition unit 14 dstores the solid numbers of all particles in the storage unit 13 so asto be associated with the time step t_(ts).

FIGS. 8 to 10 are diagrams illustrating an example of the processperformed by the simulation device according to the embodiment. Asillustrated in FIG. 8, the definition unit 14 d traces particles 60 inthe radius h to recognize the particles 60 belonging to the same solid,using the above-mentioned process. FIG. 9 illustrates a linked listindicating the connection relation between particles when the particles60 in the radius h are connected.

In the example illustrated in FIG. 10, the solid numbers of all solidparticles 71 belonging to the solid to which liquid particles 70, whichhave been newly melted at the time step t_(ts), belonged as solidparticles at the time step (t_(ts)−1) are undefined and a new solidnumber is defined. In the example illustrated in FIG. 10, the solidnumber of each solid particle 71 is defined as ‘81’ which is the solidnumber of a solid 81 or ‘82’ which is the solid number of a solid 82. Inthe example illustrated in FIG. 10, since the shortest distance betweenthe particle 71 in the solid 81 and the particle 71 in the solid 82 is r(>h), the particles 71 are defined such that they do not belong to onesolid but belong to any one of two solids.

The display control unit 14 e controls the display of various kinds ofinformation. An aspect of the display control unit 14 e will bedescribed. For example, when the definition unit 14 d stores the solidnumbers of all particles in the storage unit 13 so as to be associatedwith the time step t_(ts), the display control unit 14 e determineswhether the value of the time step t_(ts) is equal to or less than thelast time step N_(L) of the simulation. When the value of the time stept_(ts) is not equal to or less than the last time step N_(L) of thesimulation, the display control unit 14 e performs the next process.That is, the display control unit 14 e acquires the position, velocity,density, internal energy, state, and solid numbers of all particles,which are stored in the storage unit 13 of each time step, at all timesteps. Then, the display control unit 14 e controls the display of thedisplay unit 12 such that the simulation result (the position, velocity,density, internal energy, state, and solid numbers of all particles atall time steps) is displayed.

The control unit 14 is a hard-wired logic, such as an applicationspecific integrated circuit (ASIC) or a field programmable gate array(FPGA). Alternatively, a central processing unit (CPU) or a microprocessing unit (MPU) executes a program to implement the function ofthe control unit 14.

Flow of Process

Next, the flow of the process performed by the simulation device 10according to this embodiment will be described. FIGS. 11A and 11B areflowcharts illustrating the procedure of a simulation process accordingto the embodiment. The simulation process is performed at various times.For example, when a simulation execution instruction to perform thesimulation process is input from the input unit 11, the simulationprocess is performed by the control unit 14.

As illustrated in FIGS. 11A and 11B, the calculation unit 14 a sets thevalue of the time step t_(ts) to 0 (S101). Then, the calculation unit 14a determines whether an initial value of each particle is input from theinput unit 11 (S102). When the initial value is not input (No in stepS102), the calculation unit 14 a performs the determination in S102again. On the other hand, when the initial value is input (Yes in stepS102), the calculation unit 14 a increases the value of the time stept_(ts) by 1 (S103).

Then, the calculation unit 14 a calculates the position, velocity,density, and internal energy of all particles, which have been in theliquid state at the time step (t_(ts)−1), at the time step t_(ts) usingthe above-mentioned Expressions (1) to (4) (S104). Then, the update unit14 b updates the position, velocity, density, and internal energy of allparticles which have been in the liquid state at the time step(t_(ts)−1) to the calculated position, velocity, density, and internalenergy at the time step t_(ts) (S105).

Then, the calculation unit 14 a performs time evolution calculationusing the equation of motion of a rigid body for all solids includingthe particles which have been in the solid state at the time step(t_(ts)−1) and performs the next process. That is, the calculation unit14 a calculates the position, velocity, density, and internal energy ofall particles, which have been in the solid state at the time step(t_(ts)−1), at the time step t_(ts) (S106).

Then, the update unit 14 b updates the position, velocity, density, andinternal energy of all particles which have been in the solid state atthe time step (t_(ts)−1) to the calculated position, velocity, density,and internal energy at the time step t_(ts), respectively (S107).

Then, the update unit 14 b updates the state of all particles on thebasis of the calculated internal energy (S108). Then, the update unit 14b stores the update result (the position, velocity, density, internalenergy, and state) of all particles in a predetermined area of thestorage unit 13 so as to be associated with the time step t_(ts) (S109).

The determination unit 14 c determines whether there is a newlysolidified particle on the basis of the state of all particles at thetime step t_(ts) before and after the update (S110). When there is nonewly solidified particle (No in step S110), the process proceeds toS113.

On the other hand, when there is a newly solidified particle (Yes instep S110), the determination unit 14 c specifies the newly solidifiedparticle (S111). Then, the determination unit 14 c sets the solid numberof the specified solid particle to an undefined state (S112).

Then, the determination unit 14 c determines whether there is a newlymolten liquid particle on the basis of the state of all particles at thetime step t_(ts) before and after the update (S113). When there is nonewly molten liquid particle (No in step S113), the process proceeds toS116. On the other hand, when there is a newly molten liquid particle(Yes in step S113), the determination unit 14 c specifies the newlymolten liquid particle (S114). Then, the determination unit 14 c setsthe solid numbers of all solid particles belonging to the solidincluding the specified liquid particle which has been a solid particleat the time step (t_(ts)−1) to the undefined state (S115).

Then, the determination unit 14 c determines whether there is a newlysolidified particle and whether there is a newly molten liquid particleagain, on the basis of the state of all particles at the time stept_(ts) before and after the update (S116).

When there is no newly solidified particle and no newly molten liquidparticle (No in step S116), the process proceeds to S119. On the otherhand, when there is a newly solidified particle or there is a newlymolten liquid particle (yes in step S116), the definition unit 14 dperforms the next process. That is, the definition unit 14 d performs asolid number definition process (S117). Then, the definition unit 14 dstores the solid numbers of all particles in the storage unit 13 so asto be associated with the time step t_(ts) (S118).

Then, the display control unit 14 e determines whether the value of thetime step t_(ts) is equal to or less than the last time step N_(L) ofthe simulation (S119). When the value of the time step t_(ts) is equalto or less than the last time step N_(L) of the simulation (Yes in stepS119), the process returns to S103. On the other hand, when the value ofthe time step t_(ts) is not equal to or less than the last time stepN_(L) of the simulation (No in step S119), the display control unit 14 eperforms the next process. That is, the display control unit 14 eacquires the position, velocity, density, internal energy, state, andsolid numbers of all particles, which are stored in the storage unit 13so as to be associated with each time step, at all time steps. Then, thedisplay control unit 14 e controls the display of the display unit 12such that the simulation result (the position, velocity, density,internal energy, state, and solid numbers of all particles at all timesteps) is displayed (S120) and the process ends.

FIG. 12 is a flowchart illustrating the procedure of the solid numberdefinition process according to the embodiment. As illustrated in FIG.12, the definition unit 14 d defines whether there is a solid particlewhich has not been selected among the solid particles with the undefinedsolid numbers (S201).

When there is a solid particle which has not been selected (Yes in stepS201), the definition unit 14 d selects one of the solid particles whichhave not been selected and have the undefined solid number (S202). Then,the definition unit 14 d determines whether a solid particle belongingto a solid is present in a sphere with a radius h which has the selectedsolid particle as its center (S203).

When a solid particle is present in the sphere (Yes in step S203), thedefinition unit 14 d determines whether the solid particle which isdetermined to be present belongs to each of a plurality of solids(S204).

When the solid particle which is determined to be present does notbelong to any of the plurality of solids (No in step S204), thedefinition unit 14 d sets the value of the solid number of the solidparticle in the sphere to the solid number of the selected solidparticle (S205) and the process returns to S201.

On the other hand, when the solid particle which is determined to bepresent belongs to each of the plurality of solids (yes in step S204),the definition unit 14 d performs the next process. That is, thedefinition unit 14 d sets the value of the solid number of the solidparticle closest to the selected solid particle among the plurality ofsolid particles in the sphere to the solid number of the selected solidparticle (S206).

Then, the definition unit 14 d specifies the solid particle with a solidnumber to which a value other than the value of the solid number set tothe selected solid particle is set, among the plurality of solidparticles in the sphere (S207). Then, the definition unit 14 d updatesthe values of the solid numbers of all solid particles belonging to thesolid including the specified solid particle to the value which is setto the solid number of the selected solid particle (S208) and theprocess returns to S201.

When the solid particle belonging to the solid is absent in the spherewith the radius h which has the selected solid particle as its center(No in step S203), the definition unit 14 d sets, to the solid number ofthe selected solid particle, a value which does not overlap the valuesof the solid numbers of the other solid particles (S209) and the processreturns to S201.

When there is no solid particle which has not been selected among thesolid particles with the undefined solid numbers (No in step S201), thedefinition unit 14 d stores the processing result in the internal memoryand returns to the process.

As described above, the simulation device 10 according to thisembodiment performs the following process for a particle which is in aliquid state at the time step (t_(ts)−1) among a plurality of particleswhen metal in the liquid and solid states is represented as a pluralityof particles. That is, the simulation device 10 calculates the state ofthe liquid particle at the time step t_(ts) after the time step(t_(ts)−1). Then, the simulation device 10 determines whether the liquidparticle at the time step (t_(ts)−1) has become the solid particle atthe time step t_(ts) on the basis of the calculated state. Then, when itis determined that the liquid particle at the time step (t_(ts)−1) hasbecome the solid particle at the time step t_(ts), the simulation device10 performs the next process. That is, the simulation device 10 definesthe solid particle and all particles belonging to the solid includingthe other solid particles in the sphere from the solid particle, asparticles belonging to the same solid. Then, the simulation device 10calculates the state of each particle belonging to the same solid usingthe equation of motion of a rigid body. The simulation device 10 of thisembodiment defines the particle which has newly changed from a liquid toa solid and all particles belonging to the solid including the othersolid particles in the sphere with the radius h, which has the changedparticle as its center, as particles belonging to the same solid andperforms the next process. That is, the simulation device 10 accordingto this embodiment calculates the state of each particle belonging tothe same solid using the equation of motion of a rigid body. Thesimulation device 10 according to this embodiment defines the particlesthat belong to the same solid and are disposed in the sphere with theradius h, which has the particle as its center, as the same solid andcalculates the state of each particle. Therefore, according to thesimulation device 10 of this embodiment, even in a method which uses theequation of motion of a liquid for a liquid portion and uses theequation of motion of a rigid body for a solid portion, it is possibleto perform accurate calculation.

The simulation device 10 according to this embodiment calculates thetime evolution of a solid particle using the equation of motion of arigid body. Therefore, in comparison to the case when the viscositycoefficient of a solid particle is increased and the time evolution iscalculated, it is possible to further suppress an increase in theviscosity coefficient. According to the simulation device 10 of thisembodiment, the time step is not reduced in calculation. In addition,according to the simulation device 10 of this embodiment, it is possibleto suppress a significant increase in the number of calculationoperations until calculation ends. Therefore, according to thesimulation device 10 of this embodiment, it is possible to suppress asignificant increase in the calculation time.

According to the simulation device 10, when there are a plurality ofsolids including solid particles which are other than a newly solidifiedparticle and are within a predetermined range from the newly solidifiedparticle, the newly solidified particle and all particles belonging tothe plurality of solids are defined to as particles belonging to thesame solid. Therefore, the newly solidified particle makes it possibleto define all solid particles which belong to each of the plurality ofsolids to belong to one solid.

The simulation device 10 calculates the state of a particle which was ina solid state at the time step (t_(ts)−1) among a plurality of particlesat the time step t_(ts) after the time step (t_(ts)−1). Then, thesimulation device 10 determines whether the particle which was in asolid state at the time step (t_(ts)−1) has become a liquid particle atthe time step t_(ts), on the basis of the state of the particle at thetime step t_(ts). When it is determined that the particle which was in asolid state at the time step (t_(ts)−1) has become a liquid particle atthe time step t_(ts), the simulation device 10 performs the nextprocess. That is, the simulation device 10 defines the particle which isin a solid state at the time step (t_(ts)−1) and other particles in thesphere with the radius h which has the solid particle as its centeramong all particles belonging to the solid including the solid particleas particles belonging to the same solid. Then, the simulation device 10calculates the state of each particle belonging to the same solid usingthe equation of motion of a rigid body. Therefore, according to thesimulation device 10 of this embodiment, particles that belong to thesame solid and are arranged in the sphere with the radius h which has aparticle belonging to a solid including a newly molten liquid particleas its center are defined as the same solids and the state of eachparticle can be calculated. According to the simulation device 10 ofthis embodiment, when a particle is newly melted, a process of definingbelonging is performed for the solid particles which belong to the solidincluding the newly molten liquid particle. Therefore, according to thesimulation device 10 of this embodiment, it is possible to improve theaccuracy of defining the belonging of solid particles to the solid towhich the particle belonged before it is newly molted. As a result,according to the simulation device 10 of this embodiment, it is possibleto calculate the state of each particle with high accuracy.

The apparatus according to the embodiment of this disclosure has beendescribed above, but various other embodiments of the invention may bemade. For example, among the processes described in the embodiment, someor all of the processes which are automatically performed may bemanually performed.

Furthermore, each step described in the processes according to theembodiment may be arbitrarily divided or combined, depending on variousloads or usage conditions. In addition, the steps may be omitted.

The order of the steps described in the processes according to theembodiment may be changed, depending on various loads or usageconditions.

The drawings are conceptual diagrams illustrating the functions of eachcomponent of the apparatus, and the components are not necessarilyphysically configured as illustrated in the drawings. That is, thedetailed form of the dispersion and integration of the apparatus is notlimited to that illustrated in the drawings, but some or all of thecomponents of the apparatus may be functionally and physically dispersedand integrated in an arbitrary unit, depending on various loads or usageconditions.

Simulation Program

A computer system, such as a personal computer or a workstation,executes a program which is prepared in advance to implement thesimulation process of the simulation device 10. Next, an example of acomputer which executes a simulation program having the same function asthe simulation device 10 will be described with reference to FIG. 13.

FIG. 13 is a diagram illustrating the computer which executes thesimulation program. As illustrated in FIG. 13, a computer 300 includes acentral processing unit (CPU) 310, a read only memory (ROM) 320, a harddisk drive (HDD) 330, and a random access memory (RAM) 340. These units300 to 340 are connected to each other through a bus 350.

The HDD 330 stores in advance a simulation program 330 a whichimplements the same functions as those of the calculation unit 14 a, theupdate unit 14 b, the determination unit 14 c, the definition unit 14 d,and the display control unit 14 e. The simulation program 330 a may beappropriately separated.

The CPU 310 reads the simulation program 330 a from the HDD 330 andexecutes the simulation program 330 a.

The HDD 330 stores the metal model data stored in the storage unit 13illustrated in FIG. 2.

The CPU 310 reads data from the HDD 330 and stores the data in the RAM340. In addition, the CPU 310 executes the simulation program 330 ausing various kinds of data stored in the RAM 340. Not all of the datastored in the RAM 340 may be constantly be stored in the RAM 340.Alternatively, a portion of the data which is used in the process may bestored in the RAM 340.

The simulation program 330 a may not be stored in the HDD 330 from thebeginning.

For example, the program is stored in a ‘portable physical medium’, suchas a flexible disk (FD), a CD-ROM, a DVD disk, a magneto-optical disk,or an IC card, which is inserted into the computer 300. Then, thecomputer 300 may read the program from the portable physical medium andexecute the read program.

In addition, the program is stored in ‘another computer (or a server)’connected to the computer 300 through a public line, the Internet, aLAN, a WAN, or the like. Then, the computer 300 may read the programfrom another computer and execute the read program.

According to an aspect of an embodiment, it is possible to performcalculation with high accuracy even when a method is used which uses theequation of motion of a liquid for a liquid portion and uses theequation of motion of a rigid body for a solid portion.

All examples and conditional language recited herein are intended forpedagogical purposes of aiding the reader in understanding the inventionand the concepts contributed by the inventor to further the art, and arenot to be construed as limitations to such specifically recited examplesand conditions, nor does the organization of such examples in thespecification relate to a showing of the superiority and inferiority ofthe invention. Although the embodiments of the present invention havebeen described in detail, it should be understood that the variouschanges, substitutions, and alterations could be made hereto withoutdeparting from the spirit and scope of the invention.

What is claimed is:
 1. A computer-readable recording medium havingstored therein a simulation program causing a computer to execute aprocess comprising: calculating a state of a particle which was in aliquid state at a first time among a plurality of particles at a secondtime after the first time when a continuum including a liquid and asolid is represented by the plurality of particles; determining whetherthe particle which was in the liquid state at the first time has becomea first solid particle at the second time on the basis of the state ofthe particle at the second time; defining the first solid particle andall particles belonging to a solid which includes a second solidparticle arranged in a predetermined range from the first solid particleas particles belonging to the same solid when it is determined that theparticle which was in the liquid state at the first time has become thefirst solid particle at the second time; and calculating the state ofeach of the particles belonging to the same solid using an equation ofmotion of a rigid body.
 2. The computer-readable recording mediumaccording to claim 1, wherein, the defining defines, when the firstsolid particle and all particles belonging to the solid which includesthe second solid particle are defined as the particles belonging to thesame solid and there is a plurality of solids which include the secondsolid particle arranged in the predetermined range from the first solidparticle, the first solid particle and all particles belonging to theplurality of solids which include the second solid particle as theparticles belonging to the same solid.
 3. The computer-readablerecording medium according to claim 1, wherein the process furthercomprising: calculating a state of a particle which was in a solid stateat a third time among the plurality of particles at a fourth time afterthe third time; determining whether the particle which was in the solidstate at the third time has become a first liquid particle at the fourthtime on the basis of the state of the particle at the fourth time;defining all particles belonging to a solid including the particle whichwas in the solid state at the third time and other particles arranged ina predetermined range of each of all particles belonging to the solidincluding the particle which was in the solid state at the third time asparticles belonging to the same solid when it is determined that theparticle which was in the solid state at the third time has become thefirst liquid particle at the fourth time; and calculating the state ofeach of the particles belonging to the same solid using the equation ofmotion of the rigid body.
 4. A simulation method that causes a computerto perform: calculating a state of a particle which was in a liquidstate at a first time among a plurality of particles at a second timeafter the first time when a continuum including a liquid and a solid isrepresented by the plurality of particles; determining whether theparticle which was in the liquid state at the first time has become afirst solid particle at the second time on the basis of the state of theparticle at the second time; defining the first solid particle and allparticles belonging to a solid which includes a second solid particlearranged in a predetermined range from the first solid particle asparticles belonging to the same solid when it is determined that theparticle which was in the liquid state at the first time has become thefirst solid particle at the second time; and calculating the state ofeach of the particles belonging to the same solid using an equation ofmotion of a rigid body.
 5. A simulation device comprising: a calculationunit that calculates a state of a particle which was a liquid state at afirst time among a plurality of particles at a second time after thefirst time when a continuum including a liquid and a solid isrepresented by the plurality of particles, and calculates the state ofeach particle belonging to the defined same solid using an equation ofmotion of a rigid body; a determination unit that determines whether theparticle which was in the liquid state at the first time has become afirst solid particle at the second time on the basis of the state of theparticle at the second time; and a definition unit that defines thefirst solid particle and all particles belonging to a solid whichincludes a second solid particle arranged in a predetermined range fromthe first solid particle as particles belonging to the same solid whenit is determined that the particle which was in the liquid state at thefirst time has become the first solid particle at the second time.